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Theorem idlcl 35448
 Description: An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
idlss.1 𝐺 = (1st𝑅)
idlss.2 𝑋 = ran 𝐺
Assertion
Ref Expression
idlcl (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → 𝐴𝑋)

Proof of Theorem idlcl
StepHypRef Expression
1 idlss.1 . . 3 𝐺 = (1st𝑅)
2 idlss.2 . . 3 𝑋 = ran 𝐺
31, 2idlss 35447 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
43sselda 3918 1 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → 𝐴𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ran crn 5524  ‘cfv 6328  1st c1st 7673  RingOpscrngo 35325  Idlcidl 35438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-idl 35441 This theorem is referenced by:  idlsubcl  35454
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